Schematic [1 (Ge-Ge), 3 (Ge-Si), 2 (Si-Si)] percolation scheme for random GeySi1−y, as directly inspired from Ref. 6 (refer to Figs. 1 and 4 therein). On the right, the individual oscillators are labeled according to their standard percolation terminology, i.e., using a main term equipped with a superscript and a subscript, in reference to the considered bond-stretching and to the environment in which it takes place, with respect to both composition and length scale, respectively. A numerical labeling is also used (1–6), on the left, for more convenience, notably in a comparison with Figs. 3 and 5 . Limit ab initio frequencies calculated in diamond-Si (y ∼ 0, oblique-left hatching) and diamond-Ge (y ∼ 1, oblique-right hatching), on the one hand (open-red squares), and in zincblende-GeSi (y ∼ 0.5, crossed hatching), on the other hand (plain-blue squares), using either pure supercells or containing a unique impurity (as schematically indicated), are used for a qualitative discussion of the frequency shifts of the main Ge0.5Si0.5 Raman features induced by clustering (red straight-curved arrows) or anticlustering (blue semi-closed loops/arrows). The ab initio frequencies of the unique Ge-Ge, main Ge-Si, and Si-Si doublet in random-Ge0.5Si0.5, taken from the central curves ( = 0) in Fig. 5 , are added (plain-blue circles), for reference purpose. Globally, the same schematic code and labeling of the limit Raman frequencies is used in Fig. 5 . The -dependence of the individual fractions of oscillators, which monitor directly the Raman intensities, is expressed via the and probabilities in the body of the figure.
Representative Ge0.5Si0.5 Raman spectra taken from the literature, used to reveal the effect of local clustering on the (a) Ge-Si (data digitalized from Fig. 1 of Ref. 15 ) and (b) Si-Si (data digitalized from Fig. 1 of Ref. 19 ) fine structures, as emphasized by vertical arrows in each panel. The spectra refer to epitaxial layers grown as random alloys (bottom curves in each panel) or under the form of superlattices (upper curves in each panel), corresponding either to a moderate clustering [ = 0.64, panel (a)] or to a Ge4Si4 sequence with interface mixing [panel (b)]. The stars refer to the underlying Si substrate in each case. In panel (a), the upper spectrum is the difference Raman spectrum obtained by subtracting the Raman spectra of the random alloy (bottom curve) from that of the superlattice ( = 0.64), multiplied by 4 (as indicated). Corresponding percolation-type Raman lineshapes (thick-red curves) for the random (bottom curve, = 0) and clustered (upper curve, = 0.2) Ge0.5Si0.5 alloys are superimposed to the experimental data (thin-black ones), for comparison. The horizontal double-arrows mark significant phonon shifts with clustering.
-dependent percolation (black-thick curves) and MREI (black-thin curves) Ge0.5Si0.5 Raman lineshapes in case of clustering ( > 0) and anticlustering ( < 0), calculated by using the individual -dependent fractions of oscillators given in Fig. 1 . The Raman frequencies and phonon damping are taken constant, identical to those in the random Ge0.5Si0.5 alloy, in a crude approximation. The percolation-type Raman spectra corresponding to the selected values of 0.31, 0, and 0.31 are emphasized (red curves). A direct comparison can be made with corresponding -dependent ab initio Raman spectra reported in Fig. 5 . Special attention may be awarded to the sensitive Si-Si doublet identified by a specific labeling.
Positioning of the Si (small-green symbol) and Ge (large-yellow symbol) atoms in three selected 32-atom Ge0.5Si0.5 supercells corresponding to clustering ( = 0.31, left position) random substitution ( = 0, center position), and anticlustering ( = 0.31, right position). Identical values are obtained just by inverting the Si and Ge atoms in each supercell.
Ab initio Raman spectra obtained with the three 32-atom Ge0.5Si0.5 supercells displayed in Fig. 4 (thick curves), corresponding to local clustering ( = 0.31, top position), random substitution ( = 0, medium position),and local anticlustering ( = 0.31, bottom position). Additional ab initio Raman spectra obtained by inverting the positions of the Si and Ge atoms in each supercell (thin curves), thereby leaving the values unchanged, are shifted beneath the original curves, for comparison and identification of intrinsic trends. The ab initio frequencies of the unique Ge-Ge, main Ge-Si, and Si-Si doublet in random-Ge0.5Si0.5 are pointed out (plain-blue circles), for reference purpose. The Ge-Ge, Ge-Si, and Si-Si spectral ranges, delimited by dotted rectangles for help in the discussion, are identified based on limit ab initio frequencies when approaching full clustering (top-red arrows, open-red squares) and full anticlustering (bottom-blue arrows, plain-blue squares). Globally, the same schematic code and labeling of such limit Raman frequencies is used as in Fig. 1 . The sensitive Si-Si doublet is emphasized by using numbers (1,2), for unambiguous comparison with the corresponding percolation-type Raman features in Fig. 3 .
Ab initio Raman spectra obtained with two 64-atom zincblende GeSi supercells containing either one isolated Si impurity on the fcc Ge sublattice (top spectrum) or one isolated Ge impurity on the fcc Si sublattice (bottom spectrum), as schematically indicated. Distinct modes due to the GeSi-zincblende host matrix and to the isolated Ge and Si impurities are labeled using the same symbol/color code as in Figs. 1 and 5 , for a direct correspondence.
Ab initio values of the lattice parameter (a), bulk modulus (B), pressure derivative of bulk modulus (B′), and Raman frequency (ωR) of pure silicon, pure germanium, and zincblende GeSi, as obtained with different supercell sizes (differentiated by the number of atoms). The k-sampling is specified in each case.
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